The Menorah used in the Temple had 7 branches. Some synagogues have 7 branched menorahs as an ornamment, but these are usually not lit, or are lit with electric lights instead of candles.
The 9 branch menorah (if used) is used during Hanukkah. If a menorah is used for Hanukkah it is also called a hanukiah. 1 candle is used to light the other candles each night of the festival; light 1 the 1st night, 2 the 2nd night and so forth until they're all lit on the 8th night. Why 9 candles for only 8 nights? The candle used to light the ohters isn't counted.
Any group of candles will do, and i've seen Hanukkah menorahs and candle holders made of tin, glass, plaster, even apples or potatoes. Making a hanukiah out of interesting, unussual materials is a fun craft project for kids.
I've never heard of a 12 branched menorah.